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29 Nov 2018: LEMMA 5.2 (Dümbgen, Samworth and Schuh-macher, 2011). Let P Pd , let φ := log ψ (P ),and let P (B) := B eφ for any Borel set B Rd. ... P (B) =d. j =1Pj. (w j B. ) B Bd ,for some W = (w1,. ,

14 Jul 2018: There exists B = B(D,R, ) (0,) such that,for G′ GD,R satisfying δG′ ,. (5.4)µ(G). f(K L) β(G′) = µ(G′), if K = K(G,G′) > ... 2dn 1). eAnβn,. as required. Proposition 7.4. There exists B = B(d,r,δG) > 0, that is non-decreasing in d, r,and

3 Oct 2018: 1(x is a trifurcation).  c |B(n)| = c (2n 1)d. ... B(k,) = [, (2k 1)] [,]. and B(,) = B() = [,]2.

4 Jan 2018: N1 if B B(s, a, b, t),N1 if B B(s, b, a, t),. ... A B B. C C. r1. r′1r2. r′2 r′3. r3. A.